Project Overview

We decided to analyze Sports data in this project, specifically data from Formula 1 races. We have included a brief overview of the data as well as a data dictionary in the following sections.

Following this are sections that attempt to use various methods of regression and classification to answer some research questions about the data. The methods included are:

• Multiple Linear regression
• Ridge Regression
• LOESS, or natural cubic spline fit
• kNN classification
• Naive Bayes Classification
• Logistic Regression Classification

These methods were split equally between the partners. Nandini Bhelke worked on the first three regression methods, while Kevin Hallissey worked on the last three classification methods.

Citations

F1 Data and Images:

• https://openf1.org/
• https://www.stickpng.com/img/icons-logos-emojis/iconic-brands/formula-1-new-logo
• https://www.formula1.com/en/latest/article/i-couldnt-turn-hamilton-explains-japanese-gp-issue-that-caused-him-to-let.5gIdEGPh0hZ84ugrBDZZQT

MLR References:

• https://www.r-bloggers.com/2016/02/multiple-regression-lines-in-ggpairs/

Ridge Regression References:

• https://www.statology.org/ridge-regression-in-r/
• https://www.datacamp.com/tutorial/tutorial-ridge-lasso-elastic-net
• https://koalaverse.github.io/homlr/notebooks/06-regularized-regression.nb.html
• https://stackoverflow.com/questions/65346637/plotting-cross-validation-of-ridge-regressions-mse

Formula 1

Data Source and Processing

The data for this project was sourced from OpenF1 (https://openf1.org/). OpenF1 is a free and open-source API that provides real-time and historical Formula 1 data. The data from this website can be accessed in either JSON or CSV formats through the web browser. The appropriate CSV files were sourced from this website and processed to form a singular, complete dataset for this Project. The specific files used were:

• Drivers : Provides information about drivers for each session.
• Laps : Provides detailed information about individual laps.
• Meetings : Provides information about meetings. A meeting refers to a Grand Prix or testing weekend and usually includes multiple sessions (practice, qualifying, race, …).
• Position : Provides driver positions throughout a session, including initial placement and subsequent changes.
• Sessions : Provides information about sessions. A session refers to a distinct period of track activity during a Grand Prix or testing weekend (practice, qualifying, sprint, race, …).

These separate CSV files were then processed and combined to form our final dataset. A data dictionary with all variables is provided to the right. The complete dataset can also be accessed at this link: https://drive.google.com/file/d/1Bvc_8Os35966WgIHAyXb80IT2ddCGT2A/view?usp=drive_link

Data dictionary

Multiple Linear Regression Overview

Research Question: Can we predict the average speed trap speed over all laps in a Grand Prix Race using the speed trap speeds for the practice and qualifying laps?

For the Multiple Linear Regression (MLR), we focused on predicting speeds for a race using the practice and qualifying race speeds. Specifically, we focused on the speeds at the speed traps, which are a specific points on the track where the highest speeds are usually recorded. First, we divided the dataset into Practice/Qualifying Races and acquired the average speed trap speeds for those races. We then matched these with the average speed trap speeds for the Final Races to try to see if we could use the practice and qualifying races to predict speeds in the final races. Our original model equation was as follows:

\[ \textbf{Race Speed} = \beta_0 + \beta_1 \text{P}_1 + \beta_2 \text{P}_2 + \beta_3 \text{P}_3 + \beta_4 \text{Q}\]

where \(\text{P}_1\) corresponds to the average speed for all laps for Practice 1, \(\text{P}_2\) for Practice 2, \(\text{P}_3\) for Practice 3, and \(\text{Q}\) Qualifying laps.

After model selection using stepAIC(), our final model was:

\[\begin{align} \textbf{Race Speed} &= \beta_0 + \beta_1 \text{P}_2 + \beta_2 \text{P}_3 \\ &= 59.0857 + 0.6389 \textbf{P}_2 + 0.1901 \textbf{P}_3 \end{align}\]

The \(R^2\) value did increase after model selection, but only by a marginal amount. Therefore, we can see that the model does not account for all the variance in the actual data values. This was reflected in the \(R^2\) value of \(0.387\). Thus, we can conclude that our model only accounts for about 38.7% of the total variance in the actual values for race speeds.

The regression plots with correlations are also shown to the right with the response variable as average race speed in km/h. As seen in the plot, although there is some correlation between the variables, the variance is too high fo the model to accurately fit the data.

The assumptions of the multiple linear regression models were also checked. These plots are included to the right. The model seems to hold for all assumptions of equal variance for residuals, independence, and normality based on these plots. We also checked for multicollinearity by taking a closer look at the correlation between variables. The correlation plot is included to the right, which shows that multicollinearity was not an issue in this model since the correlations are under 0.8.

Conclusion: Although the model is valid, We can conclude that this is in fact not a good regression model, since we cannot accurately predict average race speeds based on the average speeds for practice and qualifying laps.

Ridge Regression Overview

Research Question: Can we predict the overall average lap duration using the average lap duration times of the track sections?

We decided to use Ridge Regression to try to predict the overall average lap duration using the average lap duration times at the three different sections in the track. We chose these variables for Ridge Regression due to multicollinearity, since the variables were highly correlated. This can be seen in the correlation matrix provided to the right.

The coefficients plot shows that higher choices for lambda values cause the beta values to approach zero. The minimum and optimal lambda values are also included in this plot. The Optimal Lambda plot also shows the process of picking the optimal lambda that minimizes the mean squared error.

The Model Fit summary to the right gives us the coefficients of the model using the Ridge Regression method. We can also see that the optimal lambda value is about 5.16. In order to determine the goodness of fit of this model, we used our fitted model to get the predicted y values. These were then used to calculate the sum of squared errors in order to get the \(R^2\) value, which is given below.

This means that our Ridge Regression model accounts for 86.48% of the variance in the response variable, or the average lap duration.

Conclusion: Thus, we can conclude that this is a valid model to use in predicting the overall average lap duration using average lap duration times for each sector of the track.

LOESS Overview

Research Question: Can we predict the average speed recorded at the speed traps using the average speed recorded at position 2 in the track?

We decided to fit a LOESS regression to predict the average speed recorded at the speed traps over all laps (avg_st_speed) using the average speed recorded at position 2 on the track (avg_i2_speed). The fitted LOESS plot is given to the right. As we can see, there is some variance in this model fit, and it seems to contain some outliers. In order to assess the goodness of fit of this model, we calculated the mean squared error value using the residuals, which was 959.2609. In order to put this value into perspective, we fit a simple linear regression model to the same values and calculated the mean squared error of that to see if LOESS performed any better. These MSE values are were as follows:

As we can see, the MSE value is lower for LOESS, so we can say that the LOESS model fits the data better. However, is this model even valid? To check the model validity, we made a Residuals vs Fitted plot to observe the variance os residuals, which is given to the right. Although clustered near the right side of the plot, possibly due to outliers, the residuals seem to be randomly scattered around zero, so we can say that the equal variance assumption holds. Next, we decided to check for normality, which raised some issues. Both the Normal Q-Q plot and the Shapiro Wilk Normality test were utilized, as seen to the right under the “Normality Assumption” section. The model did not pass the normality test, so we have to say that the model is actually invalid to draw any predictions from. Looking at the trends of the data based on other plots, we believe that some outliers might be interfering with the model fitting, and causing the normality to fail since the Q-Q plot shows only a few points near the origin that skew the normality.

Conclusion: This model is not a valid model to use for predicting the average speed recorded at the speed traps using the average speed recorded at position 2 in the track. In order to handle this, we can utilize methods such as Cook’s distance or the Box-Cox method to get rid of any outliers or high leverage points to get a more valid model fit.